I've wondered about this, too, PhilX. On the one hand, I've heard about things occurring in "quanta", which are discrete steps such as energy levels in electrons in an atom, or the idea that light consists of discrete photons. Also, there is the question of the limits of how well we can measure things -- to within plus or minus X amount. The speed of light in a vacuum, for example, is only known to within a specific amount of measurement uncertainty.

Is it that things in the universe are truly continuous and we are just limited in how well we can measure them, or is it that ultimately things in the universe occur in finite packages?

Difficult question and I don't think anyone can know the answer for sure. I believe reality can be seen as an infinite continuous something, which we can conceptualize as a multiverse. If you were God you could zoom "in" and "out" in it forever. (This is what I've come to understand even though I'm a "particle" guy.)

Our universe, at least as it appears to us, is however a finite part of this field, and we humans are also finite. So the universe appears as both continuous and discrete at the same time to us. Everything is still interconnected, continuous and non-separable with everything else in it, but by the very virtue, inherent circularity of being it and looking at it at the same time, we see it as more and more discrete, eventually completely discrete (discreteness is kinda the reciprocal of continuousness). The inherent finiteness also puts limitations on what we can do, so we run into Planck-lengths; I would argue that Planck-lengths may be characteristic of our universe and/or human understanding, but they aren't really fundamental.